Two lines, one crossing. The teal line is every pair with x + y = 10; the amber line is every pair with x − y = 2. They agree at exactly one point, the red (6, 4).
Up to now an equation has had one unknown, and one number made it true. But the world keeps asking two questions at once: a hot dog and a soda cost $10 together, and the hot dog costs $2 more than the soda — how much is each? Two blanks, one sentence. In this lesson a new kind of equation is born — one with two unknowns — and you'll discover that a single such equation has not one answer but a whole line of them. Pair two equations together and the line of one crosses the line of the other at a single shared point. That crossing is the answer to both questions at once.
Keep our color habit as you read. The unknowns x and y are violet. Equation one — its line — is teal. Equation two — its line — is amber. And the one pair that satisfies both, the place where the lines meet, is the solution in red.
10.4.1 Linear equations in two unknowns
You already know a linear equation in one unknown, such as 2x + 3 = 11. It has a single letter, raised to the first power, and a single number answer. Now we let in a second letter.
A linear equation in two unknowns ties two letters together with a plus, a minus, and some numbers — and nothing fancier. The cleanest example is
x + y = 10.
"Two numbers that add to ten." The general shape is
ax + by = c,
where a, b, and c are numbers and — this is the one rule — a and b are not both zero (otherwise there's no unknown left to talk about). The word linear earns its name: each unknown appears to the first power only. No x², no xy, no x hiding under a square root.
The anatomy of a linear equation in two unknowns. Both unknowns sit at the first power; the numbers in front are the coefficients a and b.
Here is the move that makes two unknowns feel friendly: fix one, and the other is pinned. Take x + y = 10. If you decide that x = 3, the equation becomes 3 + y = 10 — a one-unknown equation you can already solve — so y = 7. Choose x = 8 instead and y = 2. You're free to pick x; once you do, y has no choice.
Key idea
A linear equation in two unknowns, ax + by = c (with a, b not both zero), has both letters at the first power. Fixing one unknown turns it into an ordinary one-unknown equation that pins the other.
Watch out
These are not linear: x² + y = 5 (a square), xy = 6 (two unknowns multiplied), 1x + y = 2 (an unknown in a denominator). Linear means flat: only first powers, only added and subtracted.
10.4.2 Solutions of a linear equation in two unknowns
With one unknown, a "solution" was a single number. With two unknowns, a solution must say something about both letters at once — so it comes as a pair. We write it in order, x first, then y, inside parentheses:
(x, y) = (3, 7).
This is an ordered pair: the order matters. (3, 7) means x = 3 and y = 7 — not the other way around. A pair is a solution of an equation when plugging both numbers in makes the equation true together. Check (3, 7) in x + y = 10: 3 + 7 = 10 ✓. It works.
How many solutions are there? In Section 10.4.1 we saw we could pick any value for x and always find a matching y. So there are infinitely many solution pairs. Let's collect a few in a table for x + y = 10 (for each chosen x, take y = 10 − x):
x
0
2
4
6
10
y
10
8
6
4
0
Five of the infinitely many pairs that add to ten: (0,10), (2,8), (4,6), (6,4), (10,0).
Now plot those five pairs as points. Each pair (x, y) is an address on the plane: go right to x, then up to y. Watch what happens.
The five solution pairs from the table fall exactly on one straight line. The whole teal line x + y = 10 is the picture of all the equation's solutions — every point on it is a pair that adds to ten, and every pair that adds to ten is on it.
The dots line up. Connect them and you get a straight line — and that line is not just those five dots, it's all of them, the complete portrait of every solution. This is the heart of the lesson:
Key idea
A solution of a two-unknown linear equation is an ordered pair(x, y) that makes the equation true. There are infinitely many, and they all lie on one straight line — the equation's graph. The line is the set of all solutions.
Watch out
An ordered pair is a package deal. To test (5, 4) in x + y = 10, you must use both numbers: 5 + 4 = 9 ≠ 10, so (5, 4) is not a solution. Checking only one of the two coordinates tells you nothing.
🎮 Try itRide the line of x + y = 10
Step x up and down. Watch y = 10 − x answer back and the point slide along the teal line — every stop is a solution. Then test a pair of your own to see if it lands on the line.
choose x3
test a pair (a,5b4)
10.4.3 Systems of linear equations in two unknowns
One equation gave a whole line of answers — too many to call "the answer." To pin things down we add a second condition that must hold at the same time. Back to the snack stand:
the two prices add to 10andthey differ by 2.
Two sentences, both about the same x and y, both true at once. In symbols:
A system of two linear equations: a curly brace ties them together to say "both of these hold." The first equation is teal, the second is amber.
The curly brace { is read "and." A system of linear equations in two unknowns is two (or more) linear equations in the same unknowns, required to be true together. Each equation, on its own, is still just a line of solutions:
x + y = 10 is the teal line through (0,10) and (10,0).
x − y = 2 is the amber line through (2,0) and (0,−2) — for example (5,3) is on it since 5 − 3 = 2.
So a system is really a question about two lines drawn on the same plane. The first line collects all the pairs satisfying equation one; the second collects all the pairs satisfying equation two. The interesting question is the one Section 10.4.4 answers: which pair, if any, is on both?
Key idea
A system joins two linear equations in the same unknowns with a brace, meaning "both true at once." Geometrically it is two lines sharing one coordinate plane.
10.4.4 The solution of a system
The solution of a system is the ordered pair that satisfies every equation in it — the one set of values that makes equation one true andequation two true. On the plane, a pair is on line one when it lies on the teal line, and on line two when it lies on the amber line. To be on both, it must sit where the two lines cross.
The solution of the system is the crossing point. Both lines pass through (6, 4) and nowhere else together, so x = 6, y = 4 is the one pair that obeys both equations.
For our snack-stand system the crossing is (6, 4). We don't have to trust the picture — we can check it, the only proof that ever matters. Substitute x = 6, y = 4 into each equation:
x + y = 6 + 4 = 10
equation one ✓
x − y = 6 − 4 = 2
equation two ✓
Both hold, so (6, 4) really is the solution: the hot dog is $6, the soda is $4. (Try any other point on the teal line — say (7,3): it adds to 10 but 7 − 3 = 4 ≠ 2, so it fails equation two. Only the crossing passes both tests.)
When lines don't cross at one point
Two straight lines on a plane usually cross once — one shared point, one solution. But two other things can happen, and they have plain meanings:
Three possibilities. One crossing → exactly one solution (the usual case). Parallel → the lines never meet, so no solution. Same line → the two equations describe one line, so infinitely many solutions.
If two lines are parallel (same steepness, different heights), they never touch — the system has no solution. If the two equations are secretly the same line, every point on it works — infinitely many solutions. We'll treat these special cases fully later; for now, just know the crossing point isn't always one-and-only, even though it almost always is.
Key idea
The solution of a system is the ordered pair on both lines — the point where they cross. Always confirm it by substituting back into both equations. Two lines may instead be parallel (no solution) or identical (infinitely many).
🎮 Try itTwo lines, one crossing
Set the right-hand numbers of x + y = c₁ and x − y = c₂. The widget plots both lines, finds where they cross, and prints the solution pair. When the crossing has whole-number coordinates, the red dot lands right on a grid corner.
x + y =10
x − y =2
🎮 Try itDoes this pair satisfy both?
Pick a pair (x, y) and drop it onto the snack-stand system x + y = 10, x − y = 2. The tester checks each equation separately, then tells you whether the point is the shared solution. Only one pair turns both checks green.
x5
y3
★ The big ideas, in one breath
A linear equation in two unknowns, ax + by = c, keeps both letters at the first power; fixing one pins the other, so its solutions are ordered pairs(x, y) — infinitely many of them, lined up on a single straight line that is the picture of every solution. Stack two such equations into a system with a brace and you've drawn two lines; the solution of the system is the pair that satisfies both — the point where the lines cross — which you always confirm by substituting back into each equation. Most systems cross once; a few are parallel (no solution) or the same line (infinitely many).
Coming up next — Lesson 10.5
Reading a crossing point off a graph is quick but only as exact as your eyes. In Lesson 10.5 · Solving Two-Unknown Systems: Elimination you'll find the solution with arithmetic instead of a ruler — add or subtract the two equations to make one unknown vanish, then back-solve for the other. For x + y = 10 and x − y = 2, adding them erases y in a single line.
✎ Exercises 10.4
Work each one out first, then open the answer to check your thinking.
Which of these are linear equations in two unknowns? (a) 2x + y = 7 (b) xy = 6 (c) x² − y = 1 (d) y = 4 − 3x
Show answer
(a) and (d) are linear. Both have two unknowns, each to the first power, only added or subtracted. (b) multiplies the two unknowns together, and (c) has x² — neither is linear. (Note (d) rearranges to 3x + y = 4, the standard shape.)
In the equation x + y = 10, you decide x = 7. What must y be?
Show answer
y = 3. Substituting gives 7 + y = 10; subtract 7 from both sides, y = 3. Once x is chosen, y has no freedom left.
Is the ordered pair (4, 6) a solution of x + y = 10? Is (6, 4)? Is (3, 6)?
Show answer
(4, 6): yes — 4 + 6 = 10 ✓. (6, 4): yes — 6 + 4 = 10 ✓. (3, 6): no — 3 + 6 = 9 ≠ 10. A pair must use both coordinates to pass.
Complete a table of three solutions for 2x + y = 8, using x = 0, 1, 2.
Show answer
Solve y = 8 − 2x for each. x=0 → y=8; x=1 → y=6; x=2 → y=4. So (0, 8), (1, 6), (2, 4) — three of infinitely many, all on one line.
The pair (2, k) is a solution of 3x − y = 1. Find k.
Show answer
k = 5. Substitute x = 2: 3(2) − y = 1, so 6 − y = 1, giving y = 5. Check: 6 − 5 = 1 ✓.
Write the system "two numbers add to 12 and differ by 4" using a brace, with x the larger number.
Show answer
Add to 12: x + y = 12. Differ by 4, larger minus smaller: x − y = 4. So the system is { x + y = 12, x − y = 4 }. (You'll find its solution is (8, 4).)
Decide whether (5, 3) is the solution of the system { x + y = 8, x − y = 1 }.
Show answer
No. It passes equation one — 5 + 3 = 8 ✓ — but fails equation two: 5 − 3 = 2 ≠ 1. To be the system's solution a pair must satisfy both. (The real solution is (4.5, 3.5).)
By reading the hero graph, the lines x + y = 10 and x − y = 2 cross at (6, 4). Verify this is correct without the picture.
Show answer
Substitute x=6, y=4 into both: equation one 6 + 4 = 10 ✓; equation two 6 − 4 = 2 ✓. Both hold, so (6, 4) is indeed the solution — the graph and the algebra agree.
Find the crossing point of { x + y = 7, x − y = 1 } by reasoning about the two conditions (no formal method needed yet).
Show answer
(4, 3). Two numbers add to 7 and differ by 1. If they were equal they'd be 3.5 each; nudging up by half and down by half to make a gap of 1 gives 3.5 + 0.5 = 4 and 3.5 − 0.5 = 3. Check: 4 + 3 = 7 ✓ and 4 − 3 = 1 ✓. So (4, 3).
Without graphing carefully, explain why the system { x + y = 5, x + y = 8 } has no solution.
Show answer
No solution. The same sum x + y cannot equal both 5 and 8 — a number can't be two things at once. As lines they have the same steepness (both y = −x + b) but different heights, so they're parallel and never cross. No crossing means no shared pair.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
This lesson lays the conceptual foundation for U.S. Common Core grade 8. 8.F.A — a linear equation in two variables graphs as a straight line — is the backbone of Sections 10.4.1–10.4.2: students see that the line is the complete set of ordered-pair solutions. 8.EE.C.8a says the solution of a system corresponds to the point where the two lines intersect, which is exactly the move in Sections 10.4.3–10.4.4: a brace means "both true," and "both" geometrically means "on both lines," i.e. the crossing. 8.EE.C.8b (solving systems) is introduced here by graphing and checking, and is carried forward to elimination in Lesson 10.5.
The #1 misconception: treating an ordered pair like a one-unknown solution — checking only x (or only y), or accepting a pair that satisfies one equation of a system as the system's answer. The antidote: insist on substituting both numbers into every equation, every time. A pair is a package; "the solution of a system" must turn both checks green. Tie it to the picture — passing one equation puts you on one line; the system's answer must sit on both lines at once, and that is only the crossing point.
eastmath.com · Stage 10 · 10.4 Two Unknowns & Systems · Intuition before notation
eastmath.com · 10.4 Linear Equations in Two Unknowns and Systems · 10.4.2 Solutions of a linear equation in two unknowns